Paired kernels and their applications

TL;DR

This paper investigates paired operators and their kernels within the Hardy space, providing new insights into their structure, invariance properties, and applications to truncated Toeplitz operators.

Contribution

It introduces new results on the structure, invariance, and minimal kernels of paired operators and their applications to finite-rank asymmetric truncated Toeplitz operators.

Findings

Characterization of near-invariance properties of paired kernels

Existence of minimal Toeplitz kernels for projected paired kernels

Application to kernels of finite-rank asymmetric truncated Toeplitz operators

Abstract

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly $S^*$-invariant subspace of $H^2$, is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.